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Zbl 0626.34077
Yan, Jurang
Oscillatory property for second order linear delay differential equations.
(English)
[J] J. Math. Anal. Appl. 122, 380-384 (1987). ISSN 0022-247X

This interesting paper deals with the equation (1) $x''(t)+a(t)x(g(t))=0,$ where $a\in C[0,\infty)\to [0,\infty)$, a(t)$\not\equiv 0$ on $[t\sb 0,\infty)$ $(t\sb 0\ge 0)$; $g\in C[0,\infty)\to [0,\infty)$; $0\le g(t)\le t$, $t\ge 0$, $\lim\sb{t\to \infty}g(t)=\infty$. The function sequence $\{\alpha\sb n(t)\}$ for $n=1,2,..$. and $t\ge t\sb 0$, where $\alpha\sb 0(t)=\epsilon \int\sp{\infty}\sb{t}\frac{g(s)}{s}a(s)ds,\alpha\sb n(t)=\int\sp{\infty}\sb{t}\alpha\sp 2\sb{n-1}(s)ds+\alpha\sb 0(t),$ $n=1,2,..$. and $0<\epsilon <1$ is introduced here. Sufficient conditions for (1) to be oscillatory are formulated by the functions $\alpha\sb n(t)$. The main result of the paper extends the well-known oscillation criteria of Hille, Kneser and Opial in the ordinary differential equation case and Erbe in the delay differential equation case.
[J.Ohriska]
MSC 2000:
*34K99 Functional-differential equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: oscillation criteria

Cited in: Zbl 0901.34039 Zbl 0658.34057

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